Stockmayer s equation

As in so many things in this field, if you want to work through the arguments yourself, you cannot do better than go to Flory— see Principles of Polymer Chemistry, Chapter EX. Stockmayer s equation illustrates the point we wish to make with dazzling simplicity as f the number of branches, increases, the polydispersity decreases. Thus for values of/equal to 4, 5 and 10, the polydispersity values are 1.25, 1.20 and 1.10, respectively. Note also that for / = 2, where two independent chains are combined to form one linear molecule (Figure 5-28), the polydispersity is predicted to be 1.5. Incidentally, an analogous situation occurs in free radical polymerization when chain termination is exclusively by combination. 

Our equation (27) differs from Stockmayer s (1950) in that he defined the terms as dfj, i/Sm, where /r denotes chemical potential whereas we have employed activities. Thus, his expression differs from ours by a factor RT in the denominator. This difference has been taken care of in the formulation given here, so that our working equations are essentially identical with those of Stockmayer. Also the volume factor V, appearing in Stockmayer s equations, does not appear explicitly in equation (27), since the concentrations are here expressed in volume units. 

It is observed that Bueche s equation in combination with the g1/2 rule explains the results of this work whereas in combination with g3/2 it does not. The correction to g for polydispersity brings closer the agreement between the data and the seventh power relation. The difference of a few percent between the expected and observed slopes of 7 and 6.4 may be attributed to an undercorrection for polydispersity in this regard according to Graessley s findings current theories do not sufficiently account for the reduction in viscosity with polydispersity, whether the Beasley or the Stockmayer molecular weight distribution is employed (29). 

Equation (E.8) is Stockmayer s well known result7 and the derived later by Whitney and Burchard117. Combination of (E.9) allows the particle-scattering factor of the cross-linked as117 other relationships were Eqs. (E.14), (E.10) and products to be rewritten... 

We have also tried to assess the effects of integrating Hamilton s equations numerically. This is a rather difficult task since the exact solutions to these equations are not known. However, we can use the observed conservation of total energy and linear momentum as an indication that the equations are being integrated properly. For the Stockmayer and modified Stockmayer simulations the total energy and linear momentum were conserved to 0.05 and 0.0006%, respectively, over the 600 integration steps of the production phase of these calculations. 

It may be noted that Stockmayer s differential equation for his generating function follows immediately as a special case of eqn (4) by putting 3 / t =0 (corresponding, of course, to the... 

One problem of Jacobson and Stockmayer s interpretation is observed in the cases of very small but unstrained rings, where much higher concentrations of rings were formed than were predicted. Flory and Semiyen ( ) explained this deviation by suggesting that not only did two termini have to meet within a volume Vg in order to establish a bond, they also had to approach each other from a specified direction. This direction was specified by a solid angle fraction 6a)/4iT. They explained that this term should appear in the entropy expression for process 1 in the inverse form, i.e. 4Tr/6o). In process 2, if the chains were sufficiently long, there would be no correlation between the probability for two termini of a molecule to meet within Vg and to approach within the solid angle 6a). In this case, the term 6a)/4ir would be valid for inclusion into the entropy term for process 2 and, hence, when the entropies for processes 1 and 2 were summed, these terms would cancel (and the equation for AS(3) from Jacobson and Stockmayer s theory should be valid). However, for short chains, the probability of approach... 

Equation (6.1) relates the cross-link density p to the exposure energy at any point in the cross-linking process. At the gel point, the cross-link density is such that on the average every macromolecule carries one cross-link, as expressed by Stockmayer s rule... 

Let us start to explore some implications of Stockmayer s distribution it is another fundamental equation in polymer science and can be derived from the analytical solution of the copolymerization mechanism described in Table 2.11. Its derivation is long and tedious and not really required here it is enough to realize that it reflects the MWD and CCD of polymer made according to the copolymerization mechanism shown in Table 2.11 at a given time instant. The same considerations made for Flory s distribution apply to Stockmayer s distribution they will not be repeated here. 

Moreover, Stockmayer has established that Flory s equation 3.1-115... 

The Influence of Dispersity. For different types of heterophase polymerization and in context with dispersity as a characteristic feature, it is necessary to emphasize that n strongly depends on particle size. To describe this behavior, several approximations based on equation 10 can be found (113,114). In Figure 13, some n approximations are compared regarding their dependence on the particle size, which is given by the definitions in equations 12. The first approximation rii = is based on Stockmayer s solution for the Smith-Ewart case 3 ( 3> 1). 

Stockmayer s bivariate distribution for Unear binary copolymers can be expressed by the simple equations,... 

Integrating Eq. 5 over all chain lengths, one obtains the equation describing the CCD component of Stockmayer s distribution, independently of chain length ... 

Fig. 26. Molar mass dependence of the g factor for three pregel and one postgel fraction of end linked PS stars. A good fit was obtained with the Zimm Stockmayer equation (Eq. 69) and an exponent in Eq. (70) of fi 0.63 [95] which agrees well with Kurata s estimation with b-0.6 [129]. Reprinted with permission from [129]. Copyright [1972] American Society...

They calculated the value of k by using mathematical models and found a value of 5/s for hard spheric particles. This value was obtained in an equivalent manner many years ago for a gas composed of hard spheric molecules (31). They derived this equation from osmotic pressure measurements and showed the parallelism between these virial coefficients and those obtained from light scattering. Stockmayer et al. (32) made an... 

Stockmayer 25 subsequently developed equations relating to branched-chain polymer size distributions and gel formation, whereby branch connectors were of unspecified length and branch functionality was undefined. An equation was derived for the determination of the extent of reaction where a three-dimensional, network ( gel ) forms this relation was similar to Flory s, although it was derived using another procedure. Stockmayer likened gel formation to that of a phase transition and noted the need to consider (a) intramolecular reactions, and (b) unequal reactivity of differing functional groups. This work substantially corroborated Flory s earlier studies. 

Another solution is to use the master equation in its discrete from and to perform an exact mode analysis on the resulting Hiickel matrix arbitrarily truncated. In such a case the truncation is directly associated with the finite length of the chain which is taken into account in the calculation. In fact, this procedure, proposed by Jones and Stockmayer does not lead to a closed expression for the OACF, but to an infinite series of expressions corresponding to different truncations. This makes the comparison of the J S mddel with experiments rather lengthy. Since similar ideas can now be accounted for by closed expressions, we will not present here the detailed discussion of the JS model (for a more complete discussion, see Ref. 

The general solutiem to the Smitb-Ewart difierential difference equations for reaction systems in the steady state is most readily obtained using the locus-population generating function approach. This was first demonstrated by Stockmayer (1957) and subsequently by OToole (1965). It is convenient to introduce two new parameters s and m defined as... 

The estimation of g then enables all the quantities in equation (43) to be evaluated, and hence allows a relation for a hypothetical, linear dextran g = 1) to be obtained. By this method, Wales and coworkers found g values which could not be reconciled with theoretical values calculated from the randomly branched model of Zimm and Stockmayer. This led to dextran s being assigned a herring-bone type of structure, that is, a linear backbone having branches of uniform length distributed uniformly along the chain. 

Equation (11) is the well known Stockmayer formula for the DP of randomly crossllnked chains (11). It will be2noticed that the equation (12) for the structure factor S (q ) cem be obtained from equation (11) sluply by replacing the weight-average degree of polymerization of the prlmatry chain y by its structure factor"... 

Comparison of equations (4.27) and (4.28) shows that the Flory theory in near 0-solvents predicts a numerical coefficient (2.6) that is too large by most a factor of 2. Stockmayer (1955 1960) has therefore suggested that Flory s original equation should be arbitrarily adjusted so that... 

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